Continuous-time vertex reinforced jump processes on Galton–Watson trees

Abstract

We consider a continuous-time vertex reinforced jump process on a supercritical Galton–Watson tree. This process takes values in the set of vertices of the tree and jumps to a neighboring vertex with rate proportional to the local time at that vertex plus a constant $c$. The walk is either transient or recurrent depending on this parameter $c$. In this paper, we complete results previously obtained by Davis and Volkov [Probab. Theory Related Fields 123 (2002) 281–300, Probab. Theory Related Fields 128 (2004) 42–62] and Collevecchio [Ann. Probab. 34 (2006) 870–878, Electron. J. Probab. 14 (2009) 1936–1962] by proving that there is a unique (explicit) positive $c_{\mbox{crit}}$ such that the walk is recurrent for $c\leq c_{\mbox{crit}}$ and transient for $c>c_{\mbox{crit}}$.